Simulating elastodynamics and contact in a robust and accurate way not only benefitsdesigning realistic and intriguing animations and visual effects in computer graphics, but is also essential for industrial design, robotics, mechanical engineering analysis, etc. However, existing methods are constructed under strong assumptions that limit their application scenarios to a relatively narrow range, which also make the methods sensitive to algorithmic parameters such that extensive parameter tuning often needs to be performed for nearly each different example to obtain consistent quality results. To tackle these challenges, we propose a robust, accurate, and differentiable elastodynamics and contact simulation framework that can always reliably produce consistent quality results for any codimensional solids (volumes, shells, rods, and particles) in a wide range of material, time step size, boundary condition, and resolution settings with interpenetration-free guarantees but do not require algorithmic parameter tuning. Based on solid theoretical foundations, our methods provide controllable trade-off between efficiency and accuracy for different application scenarios. All the proposed features of our methods are thoroughly verified by performing extensive experiments and analyses including comparisons to state-of-the-art methods and ablation study on multiple design choices. Our framework frees designers from extensive parameter tuning as when traditional methods are used, enables simulating brand new phenomena that are never achieved before, and already demonstrate effectiveness in a broader range of application scenarios like robotics design and engineering analysis.